User blog:Cheetahrock63/Hypercomplex Blog: Extremely multivalued functions
Home WIP If you've played with complex numbers long enough, you might know that plenty of functions are multivalued. Analytic multivalued functions range from having two outputs per input (e.g. the square root function \sqrt {z} ) to three outputs per input (e.g. the cube root function [root| \sqrt[3 {z} ]]) to at most countably infinitely many outputs per input (e.g. the natural logarithm \log (z) ). After learning about more hypercomplex number systems, one might then wonder how multivalued functions like square roots can be generalized to a system like the quaternions. An example question would be "How many quaternions are square roots of -1?" Quaternionic square roots of -1 Now at first glance, the answer to how many square roots of -1 might seem obvious. Six— \acute{i} and its negation, and the two new basis imaginary units ( j and k ) and their negations. It's sometimes helpful to think of such square roots of -1 as "clones" of \acute{i} as the real number line adjoined with any of those square roots of one will behave exactly like the complex number system (i.e. the real numbers adjoined with any "clone of \acute{i} " is isomorphic to \mathbb {C} . Adjoin \mathbb {R} with a clone of the nilpotent dual number \acute{\varepsilon} and you'll have something isomorphic to the dual numbers \mathbb {G} and adjoin \mathbb {R} with a clone of the split-complex \acute{j} and you'll have something isomorphic to the split-complex numbers \mathbb {D} .). However upon further examination, one may find that there aren't just six quaternionic square roots of -1. There are much, MUCH more. Uncountably infinitely many and all of them together form a sphere of roots. Recall (or find) that (x + y{\acute{i}} + zj + wk)^2 \\ = (x^2 - y^2 - z^2 - w^2) + 2xy{\acute{i}} + 2xzj + 2xwk for real numbers x , y , z , and k . We need to find solutions of (x + y{\acute{i}} + zj + wk)^2 = -1 . Since -1 has an imaginary part of 0, we're really just looking for solutions of x^2 - y^2 - z^2 - w^2 = -1 . And for, say, 2xy to equal zero, either x or y or both have to be zero (we're working with the real numbers—a normed division algebra—so we don't need to fear the dreaded nontrivial zero divisor boogeymen). Same thing goes for 2xz and 2xw . We'll find that the only variable that would logically make it so the imaginary part as a whole is zero is x . So now we've got - y^2 - z^2 - w^2 = -1 . Get rid of the annoying negative signs and you'll have y^2 + z^2 + w^2 = -1 . So the sum of the squares of the \acute{i} -part, j -part, and k -part of a quaternionic square root of -1 is always 1. Le gasp. A sphere of roots. And since spheres have uncountably infinitely many points on them, there are uncountably infinitely many clones of \acute{i} in the quaternions. We'll denote an arbitrary quaternionic root of -1 " u ". As clones that behave exactly like \acute{i} , them being there makes it extremely easy to generalize single-variable real analytic complex functions to quaternions. For example, you can take the sine of a quaternion or cosine of a quaternion or factorial of a quaternion. An arbitrary quaternion x+y{\acute{i}}+zj+wk can be thought of as x+vu where v is the absolute value of the imaginary part |y{\acute{i}}+zj+wk|=\sqrt{y^2 + z^2 + w^2} and u is the signum of the imaginary part {\rm {sign}} (y{\acute{i}}+zj+wk)=\frac{y{\acute{i}}+zj+wk}{\sqrt{y^2 + z^2 + w^2}} . You'll also find that any negative real number has a sphere square roots. However, it's only the negative real numbers with that many quaternionic square roots. Zero still has one square root (itself) and all the other quaternions have only two. ---- Below is true for the quaternions. \sqrt {} : *Every negative real number has one sphere corresponding to its square roots *Zero has one square root (itself) *Every other number has two square roots \sqrt3 {} : *Every nonzero real number has a sphere and a point corresponding to its cube roots *Zero has one cube root (itself) *Every other number has three cube roots \sqrt4 {} : *Every negative real number has two spheres corresponding to its tesseract roots *Every positive real number has one sphere and two points corresponding to its tesseract roots *Zero has one tesseract root (itself) *Every other number has four tesseract roots \sqrt5 {} : *Every nonzero real number has two spheres and a point corresponding to its penteract roots *Zero has one penteract root (itself) *Every other number has five penteract roots \sqrtn {} where n is a positive odd integer: *Every nonzero real number has \frac{n-1}{2} spheres and a point corresponding to its n th roots *Zero has one n th root (itself) *Every other number has n n th roots \sqrtn {} where n is a positive even integer: *Every negative real number has \frac{n}{2} spheres corresponding to its n th roots *Every positive real number has \frac{n-2}{2} spheres and two points corresponding to its n th roots *Zero has one n th root (itself) *Every other number has n n th roots \log : *Every negative real number has \aleph_0 spheres corresponding to values of its natural logarithm *Every positive real number has \aleph_0 spheres and a point corresponding to values of its natural logarithm *Zero has no natural logarithm *Every other number has \aleph_0 values of its natural logarithm Gallery Quaternionic EMFs ( \mathbb {H} ) Split-quaternionic EMFs ( \mathbb {P} ) Ternionic EMFs ( \Theta ) Tessarine EMFs ( \mathbb {BC} ) Parabolic quaternionic EMFs ( \mathbb {PH} ) Hyperbolic quaternionic EMFs ( \mathbb {HH} ) Category:Blog posts